Early exercise premium method for pricing American options under the J-model

Abstract

This study develops a new model called J-am for pricing American options and for determining the related early exercise boundary (EEB). This model is based on a closed-form solution J-formula for pricing European options, defined in the study by Jerbi (Quantitative Finance, 15:2041–2052, 2015). The J-am pricing formula is a solution of the Black & Scholes (BS) PDE with an additional function called f as a second member and with limit conditions adapted to the American option context. The aforesaid function f represents the cash flows resulting from an early exercise of the option. This study develops the theoretical formulas of the early exercise premium value related to three American option pricing models called J-am, BS-am, and Heston-am models. These three models are based on the J-formula by Jerbi (Quantitative Finance, 15:2041–2052, 2015), BS model, and Heston (Rev Financ Stud, 6:327–343, 1993) model, respectively. This study performs a general algorithm leading to the EEB and to the American option price for the three models. After implementing the algorithms, we compare the three aforesaid models in terms of pricing and the EEB curve. In particular, we examine the equivalence between J-am and Heston-am as an extension of the equivalence studied by Jerbi (Quantitative Finance, 15:2041–2052, 2015). This equivalence is interesting since it can reduce a bi-dimensional model to an equivalent uni-dimensional model. We deduce that our model J-am exactly fits the Heston-am one for certain parameters values to be optimized and that all the theoretical results conform with the empirical studies. The required CPU time to compute the solution is significantly less in the case of the J-am model compared with to the Heston-am model.